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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{% {\bf \green EXERCISE:\ }} \newcommand{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Symplectic geometry\\[15mm] \small lecture 5: Arnold Conjecture} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 18, 2021 } \end{center} \newpage {\bf \blue Lagrangian submanifolds (reminder)} \definition Let $X\subset M$ be a submanifold in a symplectic manifold $(M,\omega)$. It is called {\bf \blue Lagrangian} if $T_x X \subset T_xM$ is Lagrangian for all $x\in X$. That is to say, $\dim X = 1/2 \dim M$ and $\omega\restrict X=0$. \theorem Let $\xi\in \Lambda^1 M$ be a 1-form, and $\Gamma_\xi\subset T^* M$ its graph, considered as a submanifold in the total space of the cotangent bundle. {\bf \red Then $\Gamma_\xi$ is Lagrangian if and only if $d\xi=0$.} \proof Let $\sigma:\; x \mapsto (x, \xi(x))$ be the standard diffeomorphism from $M$ to $\Gamma_\xi$. Consider the restriction of $\theta$ to $\Gamma_\xi$. For each $u\in T_{(x, \xi(x))}\Gamma_\xi$, the form $\theta$ takes $u$ to $\xi(d\pi(u))$. This implies that $\sigma^* \theta\restrict{\Gamma_\xi} = \xi$, hence $\sigma^* \omega\restrict{\Gamma_\xi} = d\xi$. \endproof \newpage {\bf \blue Hamiltonian vector fields (reminder)} \definition Let $v\in TM$ be a vector field on a symplectic manifold $(M, \omega)$. We say that $v$ is {\bf \blue symplectomorphic} if $\Lie_v\omega=0$, that is, if $\omega$ is invariant under the corresponding diffeomorphism flow. \remark From Cartan's formula, we have $\Lie_v\omega=d (i_v\omega)$, hence {\bf \purple $v$ is symplectomorphic if and only if the $\omega$-dual 1-form is closed.} \definition Let $v\in TM$ be a symplectomorphic vector field on a symplectic manifold $(M, \omega)$, and $\eta:= i_v\omega$ the corresponding 1-form. We say that $v$ is {\bf \blue a Hamiltonian vector field} if $i_v\omega$ is exact. Its {\bf\blue Hamiltonian} is a function $f$ such that $df= i_v\omega$. The group of {\bf \blue Hamiltonian symplectomorphisms} is generated by diffeomorphisms obtained by exponents of a time-dependent vector field $v_t$, which is Hamiltonian for all $t\in [0,1]$. \remark We have an exact sequence \[ 0 \arrow \R \arrow C^\infty(M) \stackrel \delta \arrow \Ham(M) \arrow 0 \] If we identify $\Ham(M)$ with exact 1-forms, the differential $\delta:\; C^\infty(M)\arrow \Ham(M)$ {\bf \purple is identified with the de Rham differential.} \newpage {\bf \blue Hamiltonian vector fields on Lagrangian fibrations (reminder)} \definition Let $(M, \omega)$ be a symplectic manifold, and $\pi:\; M \arrow X$ a smooth submersion. It is called {\bf \blue a Lagrangian fibration} if all its fibers are Lagrangian. \claim Let $\pi:\; M \arrow X$ be a Lagrangian fibration, and $H$ a function on $X$. Then {\bf \red the corresponding Hamiltonian vector field $v$ is tangent to the fibers of $\pi$.} Moreover, {\bf \red $v$ is non-degenerate everywhere on a fiber $\pi^{-1}(x)$ if and only if $dH\neq 0$ in $x$;} otherwise $v\restrict \pi^{-1}(x)=0$. \proof Let $L:= \pi^{-1}(x)$. Consider $\omega$ as a map from $T_m M$ to $T_m^* M$. Then $\omega^{-1}$ takes 1-forms vanishing in $T_m L$ to vectors $v\in T_m M$ such that $\omega(v, \cdot)$ vanishes on $T_mL$. However, $T_mL^\bot= T_mL$ because $L$ is Lagrangian. {\bf \purple Therefore, $\omega^{-1}$ takes 1-forms vanishing on $T L$ to the vector fields tangent to $L$.} The forms vanishing on $TL$ are generated by $\pi^* \Lambda^1 X$, hence the corresponding Hamiltonian vector fields are tangent to the fibers of $\pi$. \endproof \remark Let $L= \pi^{-1}(x)$ be a fiber of a Lagrangian fibration. Then $\omega$ defines a non-degenerate pairing between $T_mL$ and $T_x X$. This implies that {\bf\red the bundle $TL$ is trivial.} \newpage {\bf \blue Hamiltonian isotopy of Lagrangian submanifolds} Let $L_0, L_1$ be Lagrangian submanifolds in $(M, \omega)$. We say that $L_0$ and $L_1$ are {\bf \blue Hamiltonian isotopic} if there exists a flow $\Psi_t$ of Hamiltonian symplectomorphisms, $\Psi_0 = \Id$, $t\in [0,1]$, such that $\Psi_1$ maps $L_1$ to $L_0$. \example Let $\omega$ be the standard symplectic form on the total space $T^*M$, and $\pi:\; T^*M \arrow M$ the corresponding Lagrangian projection. Consider an exact 1-form $\eta$ on $M$, and let $L_\eta\subset T^*M$ be its graph, considered as a Lagrangian submanifold. Let $H\in C^\infty M$, with $\eta=dH$. {\bf \purple The Hamiltonian vector field $v$ associated with $H$ is tangent to fibers of $\pi$ and acts as a translation along each fiber.} Evaluating $v$ at a fiber $\pi^{-1}(x)=T^*_x M$, we obtain that $v\restrict x = \eta \restrict x$, because $\omega(v, \cdot)=\eta$. Then $e^v= \eta$, hence {\bf \purple the Hamiltonian flow associated with $H$ takes the zero section of $T^*M$ to $L_\eta$. } \newpage {\bf \blue Darboux' theorem} \theorem {\bf \red A symplectic manifold is locally symplectomorphic to a symplectic ball} (in a neighbourhood of each point). \pstep It is sufficient to check that for any symplectic form $\omega_1$ on $\R^n$ there exists a neighbourhood $U\ni 0$ such that $(U, \omega_1)$ is symplectomorphic to a symplectic ball. {\bf \green Step 2:} Choose coordinates $x_i, y_i$ on $\R^{2n}$ in such a way that $\omega_1\restrict{T_0\R^{2n}}=\omega_0\restrict{T_0\R^{2n}}$, where $\omega_0=\sum_i dx_i \wedge dy_i$. The form $\omega_t:=t\omega_1+(1-t)\omega_0$ is non-degenerate in 0, because $\omega_1 \restrict 0=\omega_0\restrict 0$. {\bf \purple Choose a starlike neighborhood $U\ni 0$ such that $\omega_t$ is non-degenerate for all $t\in [0,1]$.} {\bf \green Step 3:} In $U$ the forms $\omega_t$ are all non-degenerate and cohomologous. As in the proof of Moser's lemma, {\bf \purple choose $\eta_t$ such that $\frac {d\omega_t}{dt}=d \eta_t$, and a vector field $v_t:=-\omega_t^{-1}(\eta_t)$, vanishing in 0.} Substracting from $\eta_t$ a constant 1-form, we may assume that $\eta_t \restrict {T_0U}=0$. Then the the coefficients of the form $\eta_t$ grow as $o(r)$, where $r$ is the distance from zero. Therefore, for $U$ sufficiently small, {\bf \purple the vector field $\Psi_t$ integrates in the whole $U$, and defines a diffeomorphism $\Psi$ between $(U, \omega_0)$ and $(\Psi(U), \omega_1)$.} Finally, since $v_t=0$ in 0, the set $\Psi(U)$ contains 0. \endproof \newpage {\bf \blue Weinstein neighbourhood theorem} The following result is proven in the same way as the Darboux' theorem. \theorem Let $X\subset M$ be a compact Lagrangian submanifold in $(M, \omega)$. Then {\bf \red there exists a neighbourhood $U$ of $X\subset M$ which is symplectomorphic to a neighbourhood of $X$ in $X\subset T^* X$.} \pstep Consider a smooth retraction (say, orthogonal projection) $\pi:\; U\arrow X$. Since $X$ is Lagrangian, $\omega$ induces a non-degenerate pairing between $TX$ and the fiberwise tangent bundle $T_\pi U$. This gives a natural isomorphism $T_\pi U\restrict X\cong T^* X$. Using the fiberwise exponent map, we obtain a diffeomorphism $\Psi$ between $U$ and a neighbourhood of zero in $T_\pi U\restrict X= T^*X$. {\bf \purple This diffeomorphism is compatible with the symplectic structure on $TU\restrict X$.} {\bf \green Step 2:} Let $\omega_0$ be the symplectic structure on $U$ induced from the embedding $U \arrow T^*X$, and $\omega_1$ the symplectic structure induced from $M$. Consider the form $\omega_t:=t\omega_1+(1-t)\omega_0$, where $t\in [0,1]$. Since $\omega_0\restrict{TU\restrict X}=\omega_1\restrict{TU\restrict X}$, {\bf \purple in a sufficiently small neighbourhood of $X$ all $\omega_t$ are non-degenerate.} Shrinking $U$ if necessarily, we may assume that this is true on all $U$. \newpage {\bf \blue Weinstein neighbourhood theorem (2)} \theorem Let $X\subset M$ be a compact Lagrangian submanifold in $(M, \omega)$. Then {\bf \red there exists a neighbourhood $U$ of $X\subset M$ which is symplectomorphic to a neighbourhood of $X$ in $X\subset T^* X$.} {\bf \green Step 3:} Since $U$ is diffeomorphic to $X$, and $X$ is Lagrangian, all $\omega_t$ are exact. Therefore we may choose a smooth family $\eta_t\in \Lambda^1 U$ of 1-forms such that $d\eta_t=\omega_t$. Denote by $j:\; X \arrow U$ the tautological embedding. Since $j^*(\omega_t)=0$, one has $d(j^*\eta_t)=0$. Replacing $\eta_t$ by $\pi^*(\text{closed 1-form})$ if necessarily, we may assume that $j^*\eta_t$ is exact. Let $f_t\in C^\infty X$ be a family of functions which satisfy $df_t =j^*\eta_t$. Replacing $\eta_t$ by $\eta_t - d(\pi^*f_t)$, {\bf \purple we obtain a family of 1-forms $\eta_t$ which satisfy $d\eta_t = \omega_t$ and $j^*\eta_t=0$.} {\bf \green Step 4:} Let $v_t:= \omega_t^{-1}(\eta_t)$, and let $\Psi_t$ be the corresponding diffeomorphism flow. Using Moser isotopy argument, we obtain that $\Psi_t^*\omega_0=\omega_t$. However, $\omega_t^{-1}$ maps the kernel of the restiction map $\Lambda^1(U)\restrict X\arrow \Lambda^1X$ to $TX\subset TU\restrict X$, because $X$ is Lagrangian. Therefore, $v_t$ preserves $X_t$. Putting a metric on $U$, we may decompose the tangent bundle as $TU=T_\pi U\oplus T_\pi U^\bot$. Let $v_t^\pi$ be the part of $v_t$ which lies in $T_\pi U$. Since $v_t^\pi$ vanishes in $X$, we have $|v_t^\pi|=o(r)$, where $r$ is the distance to $X$. Using this estimate, it is easy to see that {\bf \purple $v_t$ integrates to a diffeomorphism flow in a sufficiently small neighbourhood of $X$ and maps it to another neighbourhood of $X$.} \endproof \newpage {\bf \blue The flux of isotopic Lagrangian submanifolds} \definition Let $X\subset M$ be a compact Lagrangian submanifold, and $U\supset X$ its Weinstein neighbourhood, identified with an open subset of $T^*X$. We say that a sequence $X_i\subset U$ of Lagrangian submanifolds {\bf \blue converges to $X$ in $C^\infty$-topology} if each $X_i$ is given as a graph of a closed 1-form $\eta_i$, and $\eta_i$ converge to zero with all derivatives. \definition Define {\bf \blue the flux} on the space of Lagrangian submanifolds with $C^\infty$-topology as follows. Consider two isotopic (bot not necessarily Hamiltonian isotopic) submanifolds $L_0, L_1$, a homology class $[u]\in H_1(M,\Z)$, and homotopic circles $u_0\subset L_0$ an $u_1\subset L_1$ representing $u$, denote by $\psi:\; S^1\times [0,1]\arrow M$ the homotopy map. Then $\Flux_u(L_0, L_1):= \int_{S^1\times [0,1]}\psi^*\omega$. Let $\Flux(L_0, L_1)$ be the associated map from $H_1(M, \Z)$ to $\R$. \newpage {\bf \blue Flux: correctness of the definition} \claim Suppose that $\omega$ is exact. Then the map $\Flux_u(L_0, L_1)$ {\bf \purple is independent from the choice of representatives $u_0, u_1$ and the homotopy $u_t$.} \proof Consider a class of the cohomology of the pair $H^2(M, L_1\cup L_2)$ represented by $\omega$, and let $\delta:\; H^1(L_1\cup L_2) \arrow H^2(M, L_1\cup L_2)$ be the coboundary map of the exact sequence \[ ...\arrow H^1(L_1\cup L_2) \stackrel \delta \arrow H^2(M, L_1\cup L_2) \arrow H^2(M) \arrow H^2(L_1\cup L_2) \arrow ... \] Since $\omega$ is exact, $\omega=\delta(\rho)$, and flux is obtained as $\Flux_u(L_0, L_1)= \int_{u_0\cup -u_1}\rho$. \endproof \theorem Let $L_0,L_1$ be Lagrangian submanifolds in $(M,\omega)$. Suppose that $L_1$ is a identified with a graph of a closed 1-form $\eta$ in a Weinstein neighbourhood $U\subset L_0$. {\bf \red Then $L_1$ is Hamiltonian isotopic to $L_0$ if and only if $\Flux(L_0, L_1)$.} Moreover, {\bf \red every element of $\Hom(H_1(M, \Z), \R)$ is represented by a flux of a Lagrangian submanifold.} {\bf \green The proof follows from Claim 1 on the next slide.} \newpage {\bf \blue Hamiltonian isotopy of Lagrangian submanifolds in $T^*X$} {\bf \green Claim 1:} Let $\eta$ be a closed 1-form on $M$, and $L_\eta\subset T^*M$ be its graph, considered as a Lagrangian submanifold. {\bf \red Then $L_\eta$ is Hamiltonian isotopic to the zero section $L_0$ of $T^*M$ if and only if $\eta$ is exact.} \proof We have already obtained the Hamiltonian isotopy from exactness of $\eta$. To prove the converse implication, we need to show that $L_\eta$ is not Hamiltonian isotopic to $L_0$ when $\eta$ is not exact. {\bf \purple To prove this we compute $\Flux(L_0,L_\eta)$ and show that it is non-zero unless $\eta$ is exact. } Let $\Psi_t$ map $(x, \xi)$ to $(x, \xi+ t\eta)$, and let $v$ be the vector field tangent to this action, $v\restrict (x, \xi)=(0, \eta)$. This vector field acts on $T^*M$ by symplectomorphisms because it is locally in $M$ Hamiltonian. {\bf \purple To finish the proof, it suffices to show that $\Flux_{[u]}(L_0,L_\eta)=\int_{[u]}\eta$ for any $[u]\in H^1(M,\Z)$. } Let $u$ be its representative in the zero section of $T^* M$, with $u:\; S^1 \arrow M\subset T^* M$. The corresponding annulus $A:=\Psi_{[0,1]}(u)$ has boundaries in $u$ and $\Psi_1(u)$. Using $d\theta =\omega$, we obtain $\int_A \omega= -\int_{u}\theta + \int_{\Psi_1(u)}\theta$. The first integral vanishes, because $\theta=0$ on the zero section, and the second integral is equal to $\int_u\eta\neq 0$ because $\theta = \eta$ on $L_\eta$. We proved that $L_\eta$ is not Hamiltonian isotopic to the zero section. \endproof \newpage {\bf \blue Arnold Conjecture} \remark Let $\eta$ be an exact form, and $L_\eta\subset T^*M$ its graph, considered as a Lagrangian submanifold in $T^*M$. Denote by $L_0$ the zero section of $T^* M$. The Lagrangian submanifolds $L_\eta$ and $L_0$ intersect transversally if and only if $\eta= dH$, where $H$ is a Morse function. Clearly, {\bf \purple the number of intersection points $L_0 \cap L_\eta$ is equal to the number of critical points of a Morse function, hence $\#(L_0 \cap L_\eta) \geq \sum_i b_i(M)$.} \conjecture {\bf \blue (Arnol'd, proven by Floer)}\\ Let $L\subset T^*M$ be a Lagrangian subvariety which transversally intersects the zero section $L_0\subset T^* M$. Suppose that $L$ is Hamiltonian isotopic to $L_0$. {\bf \red Then $\#(L_0 \cap L) \geq \sum_i b_i(M)$.} \newpage {\bf \blue Strong Arnold Conjecture} \remark Note that {\bf \purple the minimal number $m(M)$ of critical points of Morse functions on $M$ is, generally speaking, strictly bigger than $\sum_i b_i(M)$.} The strong Arnold conjecture is still open. \conjecture {\bf \blue (strong Arnol'd conjecture)}\\ Let $L\subset T^*M$ be a Lagrangian subvariety which transversally intersects the zero section $L_0\subset T^* M$. Suppose that $L$ is Hamiltonian isotopic to $L_0$. {\bf \red Then $\#(L_0 \cap L) \geq m(M)$.} \conjecture {\bf \blue (Arnol'd-Givental conjecture)}\\ Let $(M, \omega)$ be a symplectic manifold, and $L_1, L_2$ Hamiltonian isotopic Lagrangian submanifolds, intersecting transversally. {\bf \red Then\\ $\#(L_1 \cap L_2) \geq \sum_i \dim H^i(M,\Z/2)$.} \end{document}